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## G = Q8×C19⋊C3order 456 = 23·3·19

### Direct product of Q8 and C19⋊C3

Aliases: Q8×C19⋊C3, C76.3C6, C192(C3×Q8), (Q8×C19)⋊3C3, C38.8(C2×C6), C4.(C2×C19⋊C3), (C4×C19⋊C3).3C2, C2.3(C22×C19⋊C3), (C2×C19⋊C3).8C22, SmallGroup(456,21)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C38 — Q8×C19⋊C3
 Chief series C1 — C19 — C38 — C2×C19⋊C3 — C4×C19⋊C3 — Q8×C19⋊C3
 Lower central C19 — C38 — Q8×C19⋊C3
 Upper central C1 — C2 — Q8

Generators and relations for Q8×C19⋊C3
G = < a,b,c,d | a4=c19=d3=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c11 >

Smallest permutation representation of Q8×C19⋊C3
On 152 points
Generators in S152
(1 58 20 39)(2 59 21 40)(3 60 22 41)(4 61 23 42)(5 62 24 43)(6 63 25 44)(7 64 26 45)(8 65 27 46)(9 66 28 47)(10 67 29 48)(11 68 30 49)(12 69 31 50)(13 70 32 51)(14 71 33 52)(15 72 34 53)(16 73 35 54)(17 74 36 55)(18 75 37 56)(19 76 38 57)(77 115 96 134)(78 116 97 135)(79 117 98 136)(80 118 99 137)(81 119 100 138)(82 120 101 139)(83 121 102 140)(84 122 103 141)(85 123 104 142)(86 124 105 143)(87 125 106 144)(88 126 107 145)(89 127 108 146)(90 128 109 147)(91 129 110 148)(92 130 111 149)(93 131 112 150)(94 132 113 151)(95 133 114 152)
(1 96 20 77)(2 97 21 78)(3 98 22 79)(4 99 23 80)(5 100 24 81)(6 101 25 82)(7 102 26 83)(8 103 27 84)(9 104 28 85)(10 105 29 86)(11 106 30 87)(12 107 31 88)(13 108 32 89)(14 109 33 90)(15 110 34 91)(16 111 35 92)(17 112 36 93)(18 113 37 94)(19 114 38 95)(39 134 58 115)(40 135 59 116)(41 136 60 117)(42 137 61 118)(43 138 62 119)(44 139 63 120)(45 140 64 121)(46 141 65 122)(47 142 66 123)(48 143 67 124)(49 144 68 125)(50 145 69 126)(51 146 70 127)(52 147 71 128)(53 148 72 129)(54 149 73 130)(55 150 74 131)(56 151 75 132)(57 152 76 133)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19)(20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38)(39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57)(58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76)(77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95)(96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114)(115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133)(134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152)
(2 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)(21 27 31)(22 34 23)(24 29 26)(25 36 37)(28 38 32)(30 33 35)(40 46 50)(41 53 42)(43 48 45)(44 55 56)(47 57 51)(49 52 54)(59 65 69)(60 72 61)(62 67 64)(63 74 75)(66 76 70)(68 71 73)(78 84 88)(79 91 80)(81 86 83)(82 93 94)(85 95 89)(87 90 92)(97 103 107)(98 110 99)(100 105 102)(101 112 113)(104 114 108)(106 109 111)(116 122 126)(117 129 118)(119 124 121)(120 131 132)(123 133 127)(125 128 130)(135 141 145)(136 148 137)(138 143 140)(139 150 151)(142 152 146)(144 147 149)

G:=sub<Sym(152)| (1,58,20,39)(2,59,21,40)(3,60,22,41)(4,61,23,42)(5,62,24,43)(6,63,25,44)(7,64,26,45)(8,65,27,46)(9,66,28,47)(10,67,29,48)(11,68,30,49)(12,69,31,50)(13,70,32,51)(14,71,33,52)(15,72,34,53)(16,73,35,54)(17,74,36,55)(18,75,37,56)(19,76,38,57)(77,115,96,134)(78,116,97,135)(79,117,98,136)(80,118,99,137)(81,119,100,138)(82,120,101,139)(83,121,102,140)(84,122,103,141)(85,123,104,142)(86,124,105,143)(87,125,106,144)(88,126,107,145)(89,127,108,146)(90,128,109,147)(91,129,110,148)(92,130,111,149)(93,131,112,150)(94,132,113,151)(95,133,114,152), (1,96,20,77)(2,97,21,78)(3,98,22,79)(4,99,23,80)(5,100,24,81)(6,101,25,82)(7,102,26,83)(8,103,27,84)(9,104,28,85)(10,105,29,86)(11,106,30,87)(12,107,31,88)(13,108,32,89)(14,109,33,90)(15,110,34,91)(16,111,35,92)(17,112,36,93)(18,113,37,94)(19,114,38,95)(39,134,58,115)(40,135,59,116)(41,136,60,117)(42,137,61,118)(43,138,62,119)(44,139,63,120)(45,140,64,121)(46,141,65,122)(47,142,66,123)(48,143,67,124)(49,144,68,125)(50,145,69,126)(51,146,70,127)(52,147,71,128)(53,148,72,129)(54,149,73,130)(55,150,74,131)(56,151,75,132)(57,152,76,133), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95)(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114)(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54)(59,65,69)(60,72,61)(62,67,64)(63,74,75)(66,76,70)(68,71,73)(78,84,88)(79,91,80)(81,86,83)(82,93,94)(85,95,89)(87,90,92)(97,103,107)(98,110,99)(100,105,102)(101,112,113)(104,114,108)(106,109,111)(116,122,126)(117,129,118)(119,124,121)(120,131,132)(123,133,127)(125,128,130)(135,141,145)(136,148,137)(138,143,140)(139,150,151)(142,152,146)(144,147,149)>;

G:=Group( (1,58,20,39)(2,59,21,40)(3,60,22,41)(4,61,23,42)(5,62,24,43)(6,63,25,44)(7,64,26,45)(8,65,27,46)(9,66,28,47)(10,67,29,48)(11,68,30,49)(12,69,31,50)(13,70,32,51)(14,71,33,52)(15,72,34,53)(16,73,35,54)(17,74,36,55)(18,75,37,56)(19,76,38,57)(77,115,96,134)(78,116,97,135)(79,117,98,136)(80,118,99,137)(81,119,100,138)(82,120,101,139)(83,121,102,140)(84,122,103,141)(85,123,104,142)(86,124,105,143)(87,125,106,144)(88,126,107,145)(89,127,108,146)(90,128,109,147)(91,129,110,148)(92,130,111,149)(93,131,112,150)(94,132,113,151)(95,133,114,152), (1,96,20,77)(2,97,21,78)(3,98,22,79)(4,99,23,80)(5,100,24,81)(6,101,25,82)(7,102,26,83)(8,103,27,84)(9,104,28,85)(10,105,29,86)(11,106,30,87)(12,107,31,88)(13,108,32,89)(14,109,33,90)(15,110,34,91)(16,111,35,92)(17,112,36,93)(18,113,37,94)(19,114,38,95)(39,134,58,115)(40,135,59,116)(41,136,60,117)(42,137,61,118)(43,138,62,119)(44,139,63,120)(45,140,64,121)(46,141,65,122)(47,142,66,123)(48,143,67,124)(49,144,68,125)(50,145,69,126)(51,146,70,127)(52,147,71,128)(53,148,72,129)(54,149,73,130)(55,150,74,131)(56,151,75,132)(57,152,76,133), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95)(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114)(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54)(59,65,69)(60,72,61)(62,67,64)(63,74,75)(66,76,70)(68,71,73)(78,84,88)(79,91,80)(81,86,83)(82,93,94)(85,95,89)(87,90,92)(97,103,107)(98,110,99)(100,105,102)(101,112,113)(104,114,108)(106,109,111)(116,122,126)(117,129,118)(119,124,121)(120,131,132)(123,133,127)(125,128,130)(135,141,145)(136,148,137)(138,143,140)(139,150,151)(142,152,146)(144,147,149) );

G=PermutationGroup([[(1,58,20,39),(2,59,21,40),(3,60,22,41),(4,61,23,42),(5,62,24,43),(6,63,25,44),(7,64,26,45),(8,65,27,46),(9,66,28,47),(10,67,29,48),(11,68,30,49),(12,69,31,50),(13,70,32,51),(14,71,33,52),(15,72,34,53),(16,73,35,54),(17,74,36,55),(18,75,37,56),(19,76,38,57),(77,115,96,134),(78,116,97,135),(79,117,98,136),(80,118,99,137),(81,119,100,138),(82,120,101,139),(83,121,102,140),(84,122,103,141),(85,123,104,142),(86,124,105,143),(87,125,106,144),(88,126,107,145),(89,127,108,146),(90,128,109,147),(91,129,110,148),(92,130,111,149),(93,131,112,150),(94,132,113,151),(95,133,114,152)], [(1,96,20,77),(2,97,21,78),(3,98,22,79),(4,99,23,80),(5,100,24,81),(6,101,25,82),(7,102,26,83),(8,103,27,84),(9,104,28,85),(10,105,29,86),(11,106,30,87),(12,107,31,88),(13,108,32,89),(14,109,33,90),(15,110,34,91),(16,111,35,92),(17,112,36,93),(18,113,37,94),(19,114,38,95),(39,134,58,115),(40,135,59,116),(41,136,60,117),(42,137,61,118),(43,138,62,119),(44,139,63,120),(45,140,64,121),(46,141,65,122),(47,142,66,123),(48,143,67,124),(49,144,68,125),(50,145,69,126),(51,146,70,127),(52,147,71,128),(53,148,72,129),(54,149,73,130),(55,150,74,131),(56,151,75,132),(57,152,76,133)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19),(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38),(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57),(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76),(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95),(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114),(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133),(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)], [(2,8,12),(3,15,4),(5,10,7),(6,17,18),(9,19,13),(11,14,16),(21,27,31),(22,34,23),(24,29,26),(25,36,37),(28,38,32),(30,33,35),(40,46,50),(41,53,42),(43,48,45),(44,55,56),(47,57,51),(49,52,54),(59,65,69),(60,72,61),(62,67,64),(63,74,75),(66,76,70),(68,71,73),(78,84,88),(79,91,80),(81,86,83),(82,93,94),(85,95,89),(87,90,92),(97,103,107),(98,110,99),(100,105,102),(101,112,113),(104,114,108),(106,109,111),(116,122,126),(117,129,118),(119,124,121),(120,131,132),(123,133,127),(125,128,130),(135,141,145),(136,148,137),(138,143,140),(139,150,151),(142,152,146),(144,147,149)]])

45 conjugacy classes

 class 1 2 3A 3B 4A 4B 4C 6A 6B 12A ··· 12F 19A ··· 19F 38A ··· 38F 76A ··· 76R order 1 2 3 3 4 4 4 6 6 12 ··· 12 19 ··· 19 38 ··· 38 76 ··· 76 size 1 1 19 19 2 2 2 19 19 38 ··· 38 3 ··· 3 3 ··· 3 6 ··· 6

45 irreducible representations

 dim 1 1 1 1 2 2 3 3 6 type + + - image C1 C2 C3 C6 Q8 C3×Q8 C19⋊C3 C2×C19⋊C3 Q8×C19⋊C3 kernel Q8×C19⋊C3 C4×C19⋊C3 Q8×C19 C76 C19⋊C3 C19 Q8 C4 C1 # reps 1 3 2 6 1 2 6 18 6

Matrix representation of Q8×C19⋊C3 in GL5(𝔽229)

 1 138 0 0 0 73 228 0 0 0 0 0 228 0 0 0 0 0 228 0 0 0 0 0 228
,
 160 99 0 0 0 93 69 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 167 0 0 0 1 46
,
 134 0 0 0 0 0 134 0 0 0 0 0 1 166 159 0 0 0 46 62 0 0 0 87 182

G:=sub<GL(5,GF(229))| [1,73,0,0,0,138,228,0,0,0,0,0,228,0,0,0,0,0,228,0,0,0,0,0,228],[160,93,0,0,0,99,69,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,167,46],[134,0,0,0,0,0,134,0,0,0,0,0,1,0,0,0,0,166,46,87,0,0,159,62,182] >;

Q8×C19⋊C3 in GAP, Magma, Sage, TeX

Q_8\times C_{19}\rtimes C_3
% in TeX

G:=Group("Q8xC19:C3");
// GroupNames label

G:=SmallGroup(456,21);
// by ID

G=gap.SmallGroup(456,21);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-19,60,141,66,1064]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^19=d^3=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^11>;
// generators/relations

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